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Advanced Functions and Pre-Calculus
This courseware extends students' experience with functions. Students will investigate the properties of polynomial, rational, exponential, logarithmic, trigonometric and radical functions; develop techniques for combining functions; broaden their understanding of rates of change; and develop facility in applying these concepts and skills. This courseware is considered prerequisite learning for the Calculus and Vectors courseware.
Units
Functions: Transformations and Properties
This unit introduces functions along with many terms and notations that will be encountered when working with them. A large portion of the unit deals with sketching functions using transformations. The unit concludes with a look at inverses.
Polynomial Functions
This unit examines key characteristics and properties of polynomial functions, supportive in determining the shape of their graphs. Focus will be placed on studying the behaviour of 3rd and 4th degree polynomial functions. Through investigation, connections will be made between the algebraic, numeric, and graphical representations of these functions.
Polynomial Equations and Inequalities
This unit develops the factoring skills necessary to solve factorable polynomial equations and inequalities in one variable. Connections are made between the real roots of a polynomial equation and the x-intercepts of the corresponding polynomial function. Skills are applied to solve problems involving polynomial functions and equations.
Rational Functions
This unit examines key characteristics and properties of rational functions, supportive in determining the shape of their graphs. Focus will be placed on studying rational functions with linear or quadratic polynomial expressions in their numerators and/or denominators. Through investigation, connections will be made between the algebraic and graphical representations of these functions.
Exponential and Logarithmic Functions
This unit examines key characteristics and properties of exponential and logarithmic functions. Techniques used to solve exponential and logarithmic equations will be taught and applied to solving problems.
Trigonometric Functions
In this unit, you will be introduced to functions whose values repeat over regular intervals. The most common such functions are called sinusoidal functions. These functions will be examined graphically and algebraically. The ultimate goal is to be able to solve realistic applications that can be modeled by this type of function.
In this module, radian measure, an alternative way to measure angles, will be defined. The connection between radians and the circle, and radians and degrees will be established. Applications involving arc length and sector area will be solved.
The unit circle will be used to make connections between angles and trigonometric ratios. Primary and reciprocal trigonometric ratios will be defined and developed further through their connection to the unit circle.
Two special right angle triangles are used in conjunction with the unit circle to determine the exact values of the six trigonometric ratios for special angles \(0\), \(\frac{\pi}{6}\), \(\frac{\pi}{4}\),\(\frac{\pi}{3}\), \(\frac{\pi}{2}\), and their integer multiples. Simple trigonometric equations will also be solved.
Using the knowledge gained from the unit circle and special triangles, the graphs of the three primary trigonometric functions \(y=\sin(x)\), \(y=\cos(x)\), and \(y=\tan(x)\) will be sketched. Properties of each of these graphs will be examined and new terminology will be introduced.
In this module, the graph of each of the reciprocal functions, \(y=\csc(x)\), \(y=\sec(x)\), and \(y=\cot(x)\), will be developed from the graph of the corresponding primary trigonometric function. Properties of these graphs will be examined.
We previously developed a general equation, \(y=af[b(x-h)]+k\), for the transformations of some function, \(y=f(x)\). In this module, we apply this work to consider transformations of \(y=\sin(x)\) and \(y=\cos(x)\). New terminology specific to sinusoidal functions will be introduced.
Given specific information about a sinusoidal function, we will develop possible equations using the sine and cosine functions. Once we have a sketch and equation, we will use both to gain more information about the function.
Many real world situations can be modeled using the sine and cosine functions. We will convert the given information to a sketch and then determine a possible trigonometric equation to model the situation. The model will be useful in answering many questions arising from the specific application.
Trigonometric Identities and Equations
This unit explores equivalent trigonometric expressions and examines strategies to prove trigonometric identities and solve a variety of trigonometric equations. Knowledge of fundamental trigonometric identities will be extended to include compound angle and double angle formulas.